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It is shown in [17] that the local duality induces a di- chotomy for the finite length modules over an artinian ring R which satisfies a polynomial identity


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VOL. 77 1998 NO. 1




The local duality L : MR 7→ RLM defined below is a useful tool both in module theory and in representation theory. For example, it is applied in [4, I, Theorem 3.9] to construct Auslander–Reiten sequences for finitely presented modules. It is shown in [17] that the local duality induces a di- chotomy for the finite length modules over an artinian ring R which satisfies a polynomial identity. The consequences of this dichotomy for the represen- tation theory of R are studied in [18] and [19]. If k is a commutative artinian ring and R a k-artin algebra, the local duality coincides on the finite length modules with the (functorial) duality D = Hom(−, E(kk)), where k is the factor ring k/ Rad k.

The local duality L : MR 7→RLM is not functorial in general. The aim of this article is to show that L has the following related properties.

• The local duality commutes with finite direct sums, up to isomorphism, provided each summand has perfect endomorphism ring (Theorem 1.6). Its relation to further dualities given by proper subrings of the endomorphism ring is investigated in Propositions 1.2 and 1.5.

• The local duality can be defined for homomorphisms f : MR → NR between R-modules and it behaves well on a class of homomorphisms which we call “endofinite” (Theorem 3.2). However, this class may not be closed under addition or composition (Examples 1 and 2).

• For artinian right pure semisimple PI-rings, the local duality induces a bijection between the isoclasses of indecomposable finite length left and right modules. We use this bijection to obtain a new proof for the fact proved by Herzog [10] that such rings are of finite representation type.

Notation. Throughout this article by a ring we mean an associative ring with an identity element. A ring R is called semilocal provided its factor R = R/ Rad R modulo the (Jacobson) radical Rad R is semisimple. We de- note by Mod-R and R-Mod the categories of all right and left R-modules,

1991 Mathematics Subject Classification: 16D90, 16G10, 16R99.



respectively. For the full subcategory of Mod-R consisting of the finite length modules we will write mod-R. Homomorphisms of modules will be written on the side of the elements which is opposite the scalars. For M, N in Mod-R the group of R-homomorphisms Hom(MR, NR) will also be denoted by (MR, NR). Obviously, (MR, NR) has a natural structure of an End NR- End MR-bimodule. Furthermore we write M ∈ NR for “M is isomorphic to a direct summand of the R-module N ”. For a right R-module M with semilocal endomorphism ring S = End MR, the local dual is defined as the left R-module

LM =RHom(SM ,SI)

of S-homomorphisms from SM to the S-injective envelope I = SE(SS) of the factor S = S/ Rad S. For the notion of purity (pure submodules, (Σ-) pure injective modules, finite matrix subgroups) we refer the reader to [11, Ch. 6–8].

1. Dualizing modules using subrings of endomorphism rings.

Suppose that M is a right R-module and has a local right perfect endomor- phism ring S = End MR. Let T be a subring of S and letTJ be an injective cogenerator of T -Mod. We denote by

LT ,JM =R(TM ,TJ )

the dual of M constructed using T and J . Proposition 1.2 is concerned with the relation between LM and LT ,JM : There exists a set X such that the sum (LM )(X) is a pure submodule of LT ,JM and LT ,JM is isomorphic to a summand of the product (LM )X. Sometimes we can obtain an isomorphism LM ∼= LT ,JM : If R is a semilocal ring whose radical factor R is an artin algebra, the isomorphism class of the dual moduleRLT ,JM does not depend on the subring T provided M has finite length both as a right R-module and as a left T -module and TJ = E(TT ) (Proposition 1.5). This extends a previous result of the author that the composition structure of the dual module does not depend on the subring T (see [17, Theorem 9]). Moreover, for a finite direct sum M of modules Mi with perfect endomorphism ring we obtainRLM ∼=L LMi (Theorem 1.6).

For the proof of Proposition 1.2 we will need the following lemma.

Lemma 1.1. Let T be a subring of a right perfect ring S, let TJ be an injective T -module and put SI0 = (TS,TJ ). There exists a set X and injective envelopes Ix of simple S-modules for x ∈ X such that `


SI0 is a pure and large submodule and SI0Q

x∈XIx. P r o o f. We decompose the socle SocSI0 = `

x∈xEx of SI0 into simple modulesSEx and write Ix for their injective envelopes. Consider the following diagram of left S-modules, where ι1, ι2 and ι3 are the canonical



SocSI0 ` Ix Q Ix


ι2 //



ι3 //


Since the functor (TS, −) : T -Mod → S-Mod preserves injective modules (and cogenerators),SI0is injective and there is f such that ι2f = ι1. This f is a monomorphism, since Im ι2 is a large submodule [1, Prop. 6.17]. Hence we have g such that f g = ι3. By Bass’ theorem, SocSI0 SI0 is a large submodule ofSI0, so g is a monomorphism and splits. Finally, ι3 is a pure monomorphism, hence so is f .

Proposition 1.2. Let T be a subring of a local right perfect ring S, let

TJ be an injective cogenerator and SMR a bimodule.

(1) LM ∈RLT ,JM .

(2) There is a set X and a pure embedding (LM )(X) RLT ,JM such that LT ,JM ∈R(LM )X.

(3) If SM is finitely generated and E(SS) is Σ-pure injective, then LT ,JM ∼=R(LM )(X) for some set X.

P r o o f. Put SI = E(SS) and SI0 = (TS,TJ ) and note that LM =

R(SM ,SI), whereas LT ,JM =R(TM ,TJ ) ∼= (SM ,SI0).

(1) We have seen in the proof of Lemma 1.1 that SI0 is an injective cogenerator, so I ∈SI0. Hence LM ∈RLT ,JM .

(2) By Lemma 1.1, we have a set X such that the sum I(X)is isomorphic to a large pure submodule ofSI0 and I0 is isomorphic to a summand of the productSIX. Hence we have

(∗) (LM )(X)= (SM ,SI)(X)⊆ (SM ,SI(X)) ⊆ (SM ,SI0) ∼=RLT ,JM and LT ,JM ∈R(SM ,SIX) = (LM )X. Since (LM )(X) R(LM )X is a pure submodule, also the embedding (LM )(X)RLT ,JM is pure.

(3) IfSI is Σ-pure injective, the pure embedding I(X) SI0 splits and we have I(X) =SI0since I(X)is large in SI0. Furthermore, if SM is finitely generated, we have equality in (SM ,SI)(X) ⊆ (SM ,SI(X)) and it follows from (∗) in (2) that LT ,JM ∼=R(LM )(X).

In several situations we can obtain an isomorphismRLM ∼= LT ,JM using the following

Lemma 1.3. LetSMRbe a bimodule, S a right perfect ring, T ⊆ S a sub- ring and TJ an injective module. If (TS,TJ ) ∼=SS, then (SM ,SE(SS)) ∼=

R(TM ,TJ ).


P r o o f. Since S is a semilocal ring, we have SocS(TS,TJ ) = (TS,TJ ) ∼=

SS and it can be easily seen (as in the proof of Lemma 1.1) that the injective modulesS(TS,TJ ) and SE(SS) are isomorphic. The claim follows from an application of the Hom-⊗-adjoint isomorphism toR(SM ,S(TS,TJ )).

The following immediate consequence is well known [5, proof of Prop.

2.7]. It shows that the dualities L and D coincide for modules over artin algebras.

Corollary 1.4. Suppose MR is a finitely generated module over a k- artin algebra R. Then RLM ∼= DM where D = (−,kE(kk)) : mod-R → R-mod is the classical duality.

The following proposition shows that the isoclass of the L-dual module does not depend on the subring.

Proposition 1.5. Let R be a semilocal ring such that R is an artin algebra. Suppose that MR is a finite length module and T ⊆ End MR a subring such that TM has finite length. Then RLM ∼= (TM ,TJ ), where

TJ = E(TT ).

P r o o f. In the first step we show that we may assume that R is an artinian PI-ring, i.e. that R is artinian and R is an artin algebra. SinceTM is finitely generated, say by m1, . . . , mt, there is a monomorphism R/A → MRt, r 7→ (m1r, . . . , mtr), where A = ann MR is the annihilator ideal. So R/A is right artinian. Since the dual module LM also has finite length as a left R-module and as a right EndTJ -module [17, Theorem 11], it follows from the same argument that R/A is also left artinian. NowRL(MR) =RL(MR/A) and also R(TMR/A,TJ ) =R(TMR,TJ ) are equal, so the claim of the first step has been shown.

In the second step we assume that MR is a finite length module over an artinian PI-ring R. Since T ⊆ End MR is a subring such that TM has finite length, we deduce from [17, Cor. 13] that T is an artinian PI-ring. By Rosenberg and Zelinsky’s theorem [15, Theorem 3] the moduleTJ is finitely generated, hence it induces a Morita duality. We claim that R(TM ,TJ ) ∼= LM . Let S = End M . The bimodule S is, both as a left T -module and as a right S-module, a finite length module over a semiprimary PI-ring. In this case the multiplicity of Se as a composition factor of the Morita dual module S(TS,TJ ) coincides with the multiplicity of eS as a composition factor of SS for each primitive idempotent e ∈ S [17, Theorem 11]. Thus (TS,TJ ) ∼=SS and the claim follows from Lemma 1.3.

Now we are able to show that the local duality commutes with finite direct sums, up to isomorphism.


Theorem 1.6. Let MR=`n

i=1Mi be a sum of modules, each with right perfect endomorphism ring. Then RLM ∼=`n


P r o o f. We may assume that MRhas the decomposition M =`t i=1Mini where the modules Mihave local perfect endomorphism ring Siand are pair- wise nonisomorphic; otherwise decompose the modules M and M1, . . . , Mn

in the theorem.

Consider S = End(M1n1⊕ . . . ⊕ Mtnt) as n × n-matrix ring and take for T the diagonal subring S1n1×. . .×Stnt, where n = n1+. . .+nt. PutSI = E(SS) andTJ = E(TT ). Now, S = Sn11×n1× . . . × Sntt×nt is a T -module satisfying

SS ∼= (TS,TJ ).

The ring S is right perfect by [1, Prop. 28.11], so it follows from Lemma 1.3 that R(SM ,SI) ∼= (TM ,TJ ). Observe that the ith factor of T acts trivially on the jth summand of M1n1 ⊕ . . . ⊕ Mtnt for 1 ≤ i, j ≤ n and j 6= i, so R(TM ,TJ ) ∼= `n

i=1(SiMi,SiE(SiSi)) = `n

i=1LMi and the claim has been shown.

2. The endomorphism ring of a homomorphism. In this section we introduce the endomorphism ring of a homomorphism f : MR→ NR as the endomorphism ring of the triple (M, N, f ) when considered as a module over the triangular matrix ring T2(R) = R R0 R, and list several properties.

Let R be a ring. Recall that a right module over T2(R) is a triple (M, N, f ) where M, N are R-modules and f : MR → NR is a homomor- phism. The ring T2(R) acts on (M, N, f ) as

(∗) (m, n) · r1 r3

0 r2

= (mr1, f (m)r3+ nr2).

Homomorphisms between T2(R)-modules (M, N, f ) and (M0, N0, f0) are those pairs of R-homomorphisms h = (µ, ν) where µ : M → M0 and ν : N → N0 satisfy f0µ = νf . We also write µ = π1(h) and ν = π2(h).

Thus the category Mod-T2(R) is equivalent to the category of homomor- phisms in Mod-R (see e.g. [7, III, Prop. 2.2] and [6]).

Definition. For a homomorphism f : MR → NR define the endomor- phism ring of f as

End f = End(M, N, f )T2(R).

We say that f is endofinite if (M, N, f ) has finite length when viewed as an End f -module. The endolength of f is the length of the left End f -module (M, N, f ). We will consider it as an element of N ∪ {∞}. Note that this length coincides with the length of the left End f -moduleπ1M ⊕π2N .


Proposition 2.1. Let R be a ring and f : MR→ NR a homomorphism.

(1) The map f :π1M → π2N is an End f -R-bimodule homomorphism.

Suppose %1 : S → End M and %2 : S → End N are ring homomorphisms such that f : %1M →%2N is an S-R-bimodule homomorphism. Then there exists a uniquely determined ring homomorphism σ : S → End f such that

%1= π1◦ σ and %2= π2◦ σ.

(2) If MR and NR are finite length modules, then End f is a semipri- mary ring.

(3) Every homomorphism in Mod-R and in R-Mod is endofinite if and only if T2(R) is an artinian ring of finite representation type.

P r o o f. (1) The proof is straightforward.

(2) The endomorphism ring of a finite length module is semiprimary (see e.g. [1, 29.3]).

(3) Recall that a ring T is artinian of finite representation type if and only if every left T -module and every right T -module is endofinite (cf. [26, Theorem 6] and [13, 11.38]).

3. The local dual of a homomorphism

Definition. Let f : M → N be a homomorphism in Mod-R and (M, N, f ) the corresponding T2(R)-module. Suppose that the endomor- phism ring S = End f is semilocal and SI = E(SS). We define the local dual of f as Lf = L(M, N, f ) = (S(M, N, f ),SI). We will consider Lf also as a homomorphism of left R-modules

Lf = (f,SI) : (π2N ,SI) → (π1M ,SI).

If R is a semilocal ring with R an artin algebra, we characterize those homomorphisms f in mod-R for which Lf is well behaved. We show in Theorem 3.2 that the following properties are equivalent: (1) f is endofinite, (2) Lf is endofinite, (3) Lf is a homomorphism between the finite length modulesRLN , andRLM , and (4) f occurs as the L-dual of a homomorphism in R-mod. However, the class of these homomorphisms may not be closed under addition (Example 1) and composition (Example 2).

The local dual of a homomorphism has the following basic properties.

Proposition 3.1. Let f : MR→ NRbe a homomorphism with semilocal endomorphism ring S. Put I =SE(SS).

(1) If M and N are finitely presented R-modules, then End Lf ∼= EndSI.

In particular , Lf has semiperfect endomorphism ring.

(2) The homomorphism f is endofinite if and only if Lf is endofinite.

Moreover , f and Lf have the same endolength in N ∪ {∞}.


(3) Assume that R is a semilocal ring with R = R/ Rad R an artin algebra. If f is endofinite, and MR and NR have finite length, then there are left R-module isomorphisms

(π1M ,SI) ∼=RLM and (π2N ,SI) ∼=RLN .

P r o o f. (1) If MRand NRare finitely presented, then so is f when con- sidered as a T2(R)-module. Hence the assertion follows from [4, I, Cor. 11.3].

(2) This is a consequence of [26, Prop. 3] applied to the T2(R)-module f . (3) If f is an endofinite homomorphism, thenπ1M andπ2N have finite length as End f -modules, so we may apply Proposition 1.5.

Definition. Suppose f : MR→ NRis a homomorphism between finitely presented modules and S = End f . If SI = E(SS) induces a Morita duality S-mod → mod-S0, where S0 = EndSI, with respect to which (M, N, f ) is reflexive, then we call f reflexive with respect to L. In this case we see from Proposition 3.1(1) that S0 = End Lf and the following diagram of S-R-bimodules commutes, where η is the evaluation map:


((π1M ,SI)S0, IS0) ((π2N ,SI)S0, IS0)

f //



LLf //

Theorem 3.2 (A dichotomy for homomorphisms). Let R be a semilocal ring whose radical factor R is an artin algebra and let f : MR → NR be a homomorphism between finite length modules. Write S = End f and I =SE(SS).

(1) Suppose f is endofinite. Then Lf is an endofinite homomorphism between the modules RLN ∼= (π2N ,SI) and RLM ∼= (π1M ,SI) of finite length. Moreover , f is L-reflexive.

(2) Suppose f is not endofinite. Then Lf is not endofinite, not both mod- ulesR(π1M ,SI) and R(π2N ,SI) have finite length, and f is not isomorphic to the L-dual or the L-bidual of a homomorphism between finite length mod- ules.

P r o o f. If R is a semilocal ring with radical factor an artin algebra, then so is the triangular matrix ring T = T2(R) [7, III, Prop. 2.5]. Since the L-dual of a right T -module (M, N, f ) is the left T -module ((π1M ,SI), (π2N,SI), Lf ), the result follows from the corresponding statement for modules [17, Theo- rem 9] and from Proposition 3.1(3).

Example 1. The sum of two endofinite homomorphisms f, f0 : MR NR may not be endofinite. Let K be a field, φ an automorphism of K such


that dimFix φK = ∞. By φK we denote the K-K-bimodule K with mul- tiplication a · b · c = φ(a)bc for a, c ∈ K and b ∈ φK. We consider the hereditary artinian PI-ring R = K K⊕φK

0 K . Note that the centre of R is the field k = Fix φ, so R is not an artin algebra; moreover, the dual- ity D = (−,kk) sends every nonzero R-module to an R-module of infinite length. We consider homomorphisms between the projective indecompos- able modules P1R = 1 00 0R and P2R = 0 00 1R. Since R is left artinian, P1and P2have finite length over their endomorphism ring, which is canon- ically isomorphic to K. Of course, the dual modules LP1 and LP2 are the indecomposable injective left R-modules, which are endofinite finite length modules.

Let f, f0 : P2→ P1 be the homomorphisms given by (0, 1) 7→ (0, (1, 0)) and (0, 1) 7→ (0, (0, 1)), respectively. Both homomorphisms have endomor- phism ring K, but the endostructure of the K-R-bimodule homomorphisms f : P2 → P1 and f : φP2 → P1 is not “compatible”. Thus, their duals are R-K-bimodule homomorphisms Lf : LP1 → LP2 and Lf0 : LP1 → (LP2)φ; however, their sum f + f0is “only” a k-R-bimodule homomorphism and the R-k-bimodule homomorphism L(f + f0) is a homomorphism between mod- ules of infinite length. In particular, irreducible morphisms between endo- finite finite length modules may or may not be endofinite.

Example 2. The composition g ◦ f of two endofinite homomorphisms f and g between indecomposable modules may not be endofinite. Let T be an infinite set, φ1 and φ2 bijections of T such that φ1 and φ2 have finite order but φ2◦ φ1 acts transitively on T . (Take e.g. T = Z, φ1(z) = −z and φ2(z) = −z + 1.) Let K = k(Xi : i ∈ T ) be the field of rational functions in variables indexed by T . We denote the k-linear action on K given by Xi7→ Xφj(i) also by φj for j = 1, 2. Since φj has finite order, the dimension of K over Fix φj is finite for j = 1, 2 [9, Theorem 3.5.5]. Since φ2◦ φ1 acts transitively on the infinite set T , we have Fix(φ2◦ φ1) = k. Put Bj= K⊕φjK for j = 1, 2 and

R =

K B1 B1⊗ B2

0 K B2

0 0 K


Then R is a hereditary artinian PI-ring. Furthermore, f : P3R → P2R and g : P2R → P1R given by (0, 0, 1) 7→ (0, 0, (1, 1)) and (0, 1, 0) 7→ (0, (1, 1), 0), respectively, are both endofinite with End f = Fix φ2 and End g = Fix φ1, but g ◦ f is not endofinite since End(g ◦ f ) = Fix(φ2◦ φ1).

4. On the pure semisimplicity conjecture. According to a theorem of Auslander [2], Ringel–Tachikawa [14] and Simson [20], an artinian ring R of finite representation type is right pure semisimple. Recall that a ring R


is said to be right pure semisimple if every right R-module is pure injective, or equivalently, if every right R-module is a direct sum of modules in ind-R, the class of finitely presented right R-modules with local endomorphism ring [11, Theorem 8.4]. It is an open question, called the pure semisimplicity conjecture (pss-conjecture), whether the converse of this result also holds.

The aim of this section is to give a new short module-theoretic proof of the pss-conjecture for left artinian polynomial identity rings.

Theorem 4.1 (Herzog). A left artinian PI-ring R is right pure semi- simple if and only if R is of finite representation type.

The pss-conjecture for artin algebras has been shown by Auslander [3];

the proof of this theorem for local PI-rings, for hereditary PI-rings and for PI-rings such that the square of the Jacobson radical is zero is due to Simson [21], [22]. For arbitrary PI-rings, the pss-theorem has been established by Herzog [10]; the result could be extended by Krause [12] to right dualizing rings, i.e. to rings for which the local dual of every finitely presented endofi- nite right R-module is finitely presented. The reader is referred to [26], [24]

and [25] for a discussion of the pure semisimplicity conjecture. In [23], [24]

and [25] also potential counterexamples in the class of hereditary rings are discussed in relation with Artin problems for division ring extensions.

Note that the assumption in Theorem 4.1 that R is left artinian can be avoided by passing to a Morita dual ring R0, which is left artinian and right pure semisimple [21, Prop. 2.4(a)]. Since our proof also collects information about the category of R-modules, we would like to avoid this change of rings.

The validity of the pss-conjecture for artin algebras is an immediate con- sequence of the existence of almost split sequences [7], and of the following proposition, due to Auslander [3, Cor. 2.3].

Proposition 4.2. Let R be a right pure semisimple ring such that there exists a left almost split morphism N → B in the category Mod-R for every module N in ind-R. Then there are only finitely many modules in ind-R, up to isomorphism.

We include a module-theoretic version of Auslander’s proof.

P r o o f. Suppose that(Mi)i∈Iis a family of pairwise nonisomorphic mod- ules in ind-R and M = Q

i∈IMi is their product. We show in three steps that the canonical pure monomorphism σ :`

i∈IMi Q

i∈IMi is an iso- morphism. Then I must be finite, and we are done.

Step 1. Each Mi occurs as a summand of M : By assumption, the sum


i∈IMi is a pure injective module, hence σ is a split monomorphism.

Step 2. Since R is right pure semisimple, M is a direct sum of modules in ind-R. We show that any direct summand N of M with N in ind-R is


isomorphic to one of the modules Mi: Assume that N is not isomorphic to any module Mi and let q : N → B be a left almost split map for N . Consider the diagram



Mi q //




By assumption on N , no πiι is a split monomorphism, so for every i ∈ I there is a map fi : B → Mi with fiq = πiι. Hence the product map f = (fi)i∈I makes the upper part of the diagram commutative, i.e. f q = ι. Since ι is a split monomorphism, so is q—a contradiction.

Step 3. Every Mi0 occurs at most once in a direct sum decomposition of M : Apply the argument in Step 2 to M0 = Q

i6=i0Mi instead of M = M0⊕ Mi0.

So for the proof of Theorem 4.1 we have to show that for every M in ind-R there exists a left almost split map M → N in the category Mod-R.

This is the case if M is an endofinite module over an artinian PI-ring R. The following lemma will be used to “transform” one chain condition on finite matrix subgroups (see [11, Prop. 6.3]) into endofiniteness.

Lemma 4.3. Let MR be a finitely presented module such that the endo- morphism ring S = End MR is right perfect. If M satisfies acc for finite matrix subgroups, then M is endofinite.

P r o o f. Since finitely generated endo-submodules of M are finite ma- trix subgroups (of type Sm1+ . . . + Smn = {f (m1, . . . , mn) : f ∈ Hom(MRn, MR)}), the module M has acc for finitely generated endo-submodules. Hence every endo-submodule of M is finitely generated. Since S is right perfect, every left S-module has dcc for cyclic submodules, hence by Bj¨ork’s theo- rem [8, Theorem 2] also dcc for finitely generated submodules. Thus M is endofinite.

Now we can give a new proof of Herzog’s Theorem 4.1.

Proof of Theorem 4.1. The ring R is right pure semisimple, so every right R-module is Σ-pure injective and hence satisfies dcc for finite matrix subgroups [26, Theorem 8.1]. Thus every left R-module has acc for finite matrix subgroups [26, Theorem 6]. Since R is left artinian, every module M ∈ R-ind has finite length. By [1, Cor. 29.3] the endomorphism ring of

RM is semiprimary, hence right perfect, so it follows from Lemma 4.3 that


M is endofinite. In particular, R is twosided artinian. Moreover, since R is a PI-ring, the transpose preserves finite endolength [18, Theorem 8], so also every module in ind-R is endofinite. But every endofinite module M in ind-R is the L-dual of a module in R-ind [17, Theorem 1] and thus there exists a left almost split map M → N in the category Mod-R [4, I, Theorem 3.9]. By Proposition 4.2, R is of finite representation type.

Acknowledgements. For helpful discussions the author would like to thank L. Angeleri H¨ugel, H. Krause and W. Zimmermann, the supervisor of the author’s doctoral dissertation [16] from which the material of Section 4 is taken.


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Katedra Algebry MFF–UK Sokolovsk´a 83

18675 Praha 8, Czech Republic E-mail: mschmidm@karlin.mff.cuni.cz

Received 17 May 1997;

revised 28 November 1997


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